Transmission line



2 Sheets-Sheet l INNER DIAMETER 0F OUTER CONDUCTOR II'I INcIIEs OPTIMUM DIAMETER RATIO R ma W .w mJ 9 m. w

BYjv- ATTORNEY Aug. 14, 1934. J. M. WEST TRANSMISSION LINE Filed May 20, 1951 2 Sheets-Sheet 2 lNVENTOR J. M WES T B) ATTORNEY Patented Aug. 14, 1934 UNITED STA TES OFFICE WWW TRANSMISSION LINE Julian West, Nuuey, J., assignor to Bell Telephone Laboratories, Incorporated, New York, NJ SL', a corporation of New York Application May-20; 1931, Serial 'No. 538,647

7 s Claims; 5(011' 178-44) The present invention relates to-a systeinfforthe transmission of'guidedelectricatsignaling waves, and more particularly to a concentric conductor line for use in a system; adapted -to transmit signals extending over a wide range of frequencies. It is an object of the invention to obtain the maximum efficiency of transmission "offwhich a concentric conductor line of fixed outer diam eteriscapable.

In another aspect 'of the invention, it isan object to obtain an optimum proportioning of the conductors in a. concentric conductor trans mission line with respect to its attenuation and torthe amount of material comprising it.

In the development 'of' the art of electrical communication it has been increasingly desirable to extend the range of frequencies transmitted over conducting lines. Lines of the openwire type have heretofore been the best available for the transmission of high frequency waves, such as used in carrier telephony. The frequent transpositions of conductors made necessary by the coupling between parallel circuits and the susceptibilityof open wire lines to interference from external sources seriously "limit" the frequency range and field of usefulness of this type of line. Cable-circuits are not well suited for use at very high frequencies because of the high capacity between conductors. a Q

The transmission element towhich the present invention relates comprises a central'c'onductor, either solid or tubular, and an outer tubular conductor concentric therewith and insulated therefrom. The dielectric between conductors is preferably gaseouayonlya suiiicient number of insulating spacersorwa'shers being provided-to maintain the conductors in their concentric relation. "A comp'oundconductor of this type has several characteristics which well adapt itto the transmission of waves of-high frequency. For example, the grounded outer conductor acts as avshield for the inner and practically eliminates.

interference from stray electric and magnetic fields.- Induction from adjacent signal conductors, for theisame. reason, isiat an extremely low level." Conversely, signals inthezconcentric conductors create but little interferenceineadjacent channels, as the fields associated with the signal waves are closelyconfined to. the annular space between central and outer conductors, Further, the efii'ciency of the line is practically independent .of weather conditions,

, as the gaseous dielectric is entirely enclosed and can be maintained at constant pressure and humidity.- Aconcentric pair of conductors havingan outer diameter of a few inches will transmit over longdistancc frequencies that can be measured in megacycles, and with only moderate attenuation. p p p I k At high frequencies the current in a concentric pair of conductors is not uniformly distributed throughout the conducting material, but tends to concentrate at the outer surface of the central conductor, and, at the inner surface of the surto: rounding conductor. The'unused portion of the material cannot be dispensed with, however, as mechanical considerations require that the con-1 ductors be made of substantial thickness. I It will befreadily appreciated that even a relatively 701 slight reduction in either attenuation or diameter of conductors will, in the aggregate, represent a considerable economy or an improvement in efficiency.

Applicant has discovered that for a pair of 76: coaxial conductors of fixed outer diameter there exists a ratio between the inner diameter of the outer conductor and the outer diameter of the inner, conductor, which, 'if observed, win result in aminimum transmission loss in waves of; a 80:, given frequency, other things being equal. This optimum ratio, ingeneral, depends on both'the diameter of the outer conductorand the fre-'' quency for which the conductor is designed"; At very hi'h frequencies, however, minimum at- 95 tenuation obtains for practically all diameters of outer conductors when the ratio of diameters issomewhat less than 3.6", similarly, if the absolute diameters of the conductors are large,f 3.6

approximately the optimum figure at all fre quencies. In a copending application of E. 1. Green and F. A. Leibe bearing Serial No. 365,- 517,, filed "May 23, 1929, it is shown that in the ranges of frequencies and diameters in which their formulae are applicable, the optimum ra tio of diameters is 3.59, independent of frequency and absolute diameters. This ratio, 3.59, applicant finds, however, is an upper asymptotic limit which the optimum ratios approach as the absolute diameters of the conductors or the fre- 10 quency, or both, approach the values at-which the equations of the Green'et a1 application supra apply. For lower values of frequency and outer diameter the expression for the optimum amiss complex function involving both factors. expression will be developed hereinafter and an explanation made of its application in anyparticu lar instance.

.'Applicant' has discovered another character isticof, awmeenua conductor system, which taken together with the discovery of an optimum diameter ratio, makes possible what is in some cases an even more efiective disposition of the material comprising the conductors. This discovery is that with a fixed outer diameter the ratio of diameters may depart considerably from that at which an absolute minimum of attenuation occurs without occasioning more than a slight change in the attenuation of the system. To provide a conductor between two points, then, which is to have a predetermined attenuation, a greater diameter ratio than the optimum, and therefore, a smaller inner conductor ,-'may ,be used, with a resulting substantial economy of copper or other material that maycomprise the conductor. w

The nature of the present invention will more fully appear from a consideration .ofthe follow: ing detailed description and accompanying drawings, of which:

.Fig. 1,shows the construction and; application of one typebf concentric conductor cable;

Fig. 2: showsgraphically the relation between" optimum diameter ratio," frequency and diameter of outer conductor; r f i Fig. 3 shows a systemembodying the present invention; and

of a concentric conductor' cable.

The bearing of diameter ratio on the attenu-j ation of a'concentric conductor cable can beseen by considering its relation to the several] .fa'cltors. determining the attenuation constant (a), viz.,'the alternatingcurrent resistance (R) of the conductors, the'capacity (C) between con-' ductors, the shunting conductance (G) {and the inductance (L) These'fac'tors are related by the equation:

Fig. 4 shows diagrammatically a cross-section Eecause of the good insulation ,betwn conductors of the concentric type, the conductance (G) is very small; the quantity containing it may usually, therefore, be disregarded. With a given diameter of outer conductor Equation (1) indicates that by increasinglt'he diameter of the central conductor, and thereby decreasing the resistance, the attenuation constant tend'sfto' be reduced. Increasing the diameter of the central conductor at the same time increases the capacity (C) between conductors; the inductance (L) is decreased. Both theseflatterchanges.

tendto increase the attenuation. ,At some point the reduction of resistance (R) attending an in crease in the diameter of the inner conductor is offset by an equal or greater increase in the factor l Q of the outer diameters of the conductors. In the application of E. I. Green et a1 supra it is stated that hiy /reg) (z) L=K11 a (3) ratio of b to a is 3.59. Within the limits of fre- "quency and diameter fixed by the approximations involvedin Equations (2 (3) and (4),.theoptimum ratio independent of frequencyand diameter. l y f f At lower frequencies skin. effect'is not as pro: nounced and cannot be assumed that the currentis confined to'the' surface layers. An equation for the attenuation constant a of more general applicability must therefore. be used.

w The following. is one of several equations accurately' expressing the propagation characteristic I of ;a concentric conductor pair:

1 ber p-Fj bei p p'log'k ber p+j bei' p J j... r r 1 Dq log k q log k] where I a=' attenuation in nepers per cm. 'li =phase shift in radians per cm.

c=3 X 10 k=b/a b=inner radius of outer conductor in cms. a=outer radius of inner conductor in ems h=thiclgess of outer conductor in ems. p=2 rvm q= w/ b )\=conductivity in abohms per cm.

- eem W A=sinh Ssin S B=sinh S+sin s Y T PS S W and ber p and bei p are Bessels functions which can, be evaluated by reference to mathematical tables, The attenuation constant is the real part of the complex-quantityon theright hand side. It canbe expressed in slightly-difierent form when some assumptions are made as to the thickness of the; conductors, Where i the quantity j i is equal to or greater than 21r,.the=conductor may be said to be electrically thick. .Provided that the outer conductor is electrically thick and the inner conductor either electrically thick or solid, the attenuation constant 'isgiven. 'by. the' expression: i z 1 1 ber p+jbi p"-.-.: p log' k ber"p +j bei' 11 4 Wh eas-. 'e rfmi e j The. following is a brief outline ofv the proof: ofEquation (5) Ina transmission line: formed of two coaxial cylindrical conductors; the dis-. tribution of. the electric force in any plane at right .anglesto .the axis is symmetrical with respect to the commonaxis of the two conductors and-is,-therefore, a function of onevariable only, namely of the distancefromthe axis. In what follows it is supposed that a'sinusoidal electromotive force is impressedupon the transmission line; Conforming. 'to the usual custom this force may be expressed ing-ha complex form Ee 'efl". It is well known that in a region occupied by; a conductor the quantity E satisfies the following differential equation I =.1/41 iwnj =2r1/fiio+j)- This is Besselfs equation and its most general solution can be expressed in terms of Bessels functions. It is also well known that the tangential components of the electric and magnetic forces are continuous across the boundary between two difierent media; similarly the normal components of the. electric and magnetic induction are also continuous.

Inside the inner conductor the particular solution of Equation ('7) subject totheabove men tioned boundary conditions is :given by the following equation V 1H 10(010] v i H v lwa) and I is the total current flowing in the inner conductor. Similarly for the outer conductor we E Q TI X U Q. where a is the inner radius of the inner conductor, and where r V ?e. f .l 2 iw;t K0(0'1)K 0(0'a 1} ab 'o( fl) "Q( l, V f r '0( .'o( )1 It can be also shown that toavery highfdegree of accuracy the electric force in a dielectric =be-' tweenthe conductors satisfies-the -following are a s- 1. M where g and Gare respectively the conductivity and the dielectric constant'of the medium between the conductors. -The' solution of Equation (12 can' be writeri as i 1.

i where A is an unknown 'constant'to bedeterminedfrom the boundary conditions; The latte 'd'equirethat the value-of-Eif computed by Equation (13) when 'r=a should equal the corresponding valueof the electric force if computed 5 by Equation (8). Similarly the valueo'f E-computed from Equation (13) for r=b should be the same as the corresponding value of Ewhen computedfrom Equation (10) These-two conditions make" possible the determination of A; moreover, they impose 'a restriction upon-I; In fact these in terms of her "and bei functionsan'd replacing the 'Bessel functions I occurring in Z2(qb,o'b;p'd') by the 'first' jitwofterms ofj'the corresponding asymptotic'expansions, I ;o'btain Equation (5) The validity of Equations (5) and (6)"is de-f pendent not on the frequency alone, flout on a function'j of frequency and j absolute @di'ameterl They afre accurateto'a fraction. ofone' per cent when q,'i.e.,"j l"' v f t V "i e is eq'ualto or' isg'reater than-l. A solution for thevalue of'k,'the ratio o'fdiameterat which the attenuation :isa minimum; canine-obtained mathematically or graphically. Itwill be'iound' by either method thatthe: optimum ratio isa' function-of both thediameter'of the outer conductorand,the'frequency-selected. i .Fig;- 2 shows graphically the variation of the optimum diameter ratiowith both frequency and absolute inner diameter of outer conductor. Frequency is scaled logarithmically alongthe axis ofordinates, diameteraratio's are indicated;by the abscissae. What curve of the family shown is to be used in designing a given concentriccon ductor pair is determined by the inner diameter of the outer conductorw Noting theintersection of this curve with the horizontal line correspond ing to the frequency selected'andobserving the abscissa of thispoint the optimum, ratio is obtained directlynv Since the attenuation increase's with frequency it is ordina'rily the highest fre quency in the'band' of frequenciesato beftransmitted that is :used in determining the diameter ratio.- ,As a numerical examplejsuppose' the highest frequency i of signals to be :transmitted over aconductor is' 1one--hundred fifty thousand cycles per second a'nclthat the conductor, perhaps one of severalwithin a' protecting sheath, is such that the inner 'diarr'ieter of the outer conductor isone-half inch.- Selecting the correspond ing'curve in Fig."2 and tracing'the: horizontal line representin'g one hundred fifty thousand cycles per second to its intersection therewith; the optimum diameter-ratio is shown on the scale of the -ab'scissae'to be 3.37. The outer diameter of the inner conductor should-thereforebe approximately fifteen hundredths l of an 4 inch. Asthefrequency isincreased and larger diam eters are used'for the outer conductorthe opti-' mum ratio likewise increases-"with 3i59-as the asymptotic limit. The-discovery of this limit of optimum valuesis 'due to E;-I.Green and RA. Leibe: and is disclosed and claimed'in their application'forpatent identified' above. 7 That discovery is not within the sco'pe of the' presenti invention, which is-directed to concentric conductors having diameter ratios lessthan' the limiting value 0F359? The optimummwill be approximately 3.55, or less; forconcentrici-conductor cables-having frequenciesand inner radii of: outer conductors .r'elated .by the empirical =ex-, pression .1; .Y b izioxioa, v wherein b is in inches and] is in cycles per conditionsrassignathefollowing: .value for. the

.In Fig. 1 is shown a pair of concentric conductors 1, 2, the outer diameter of which is limited by the size of the channels in conduit 4. The outer conductor 1 may be a hollow cylinder of copper or other suitable conducting material or a thin layer of copper on a pipe of some cheaper material .such as lead. The inner conductor 2 may be solid or tubular. It is supported in concentric relation With the outer conductor by insulating spacers 3 which are placed at frequent intervals. The spacers, or washers, are made as thin as mechanical requirements will permit and are comprised of pyrex glass or other good insulating material having a low dielectric constant and small loss angle. V

Fig. 3 shows a carrier-frequency signal trans-' mission system in which the present invention may be embodied. Two-way transmission is ob tained over the concentric conductor transmission line 14 by using separate frequency ranges for the signals transmitted in the two directions. Signals from the telephone subscribers lines 6 are transmitted through hybrid coils '7 to the individual modulators 9. The carrier waves sup- '1 plied to the modulators by oscillators 10, 20, 30,

etc., may range, for example, from twenty to eighty thousand cycles per second, the frequency of each oscillator differing from that of the next by at least the width of the band of signal frequencies. After passing through the respective band-pass'filters 11, 21, 31, etc., and amplifiers 12 the modulated carrier waves are applied to transmission line 14. The latter may be of the concentric conductor type described 1 hereinbefore. I

Signals received from. line 14 are selected by the band-pass filters 26, 36, 46, etc, passed through amplifiers 12 and. reduced to theiroriginal form in demodulators 31. The frequencies of the oscillators 29, 39, 49, etc., correspond with those of the carrier waves on which the signals are impressed. .The signals are then applied to the input terminals of hybrid coils 7 and transmitted over the subscribers lines 6. A system of this type is described in L. Espenschied Patent 1,548,260, August 4, 1925, to which reference may be made for further details of the construction ofthe apparatus and i'of the operation of the circuits employed. a

Applicant has discovered another character.- istic of coaxial conductor lines that can be used to good advantage in proportioning the conductors. It is found; that within wide limits about the optimum-diameter ratio, there is little change in attenuation with change in the diameter of the central conductor. Anexplanation of this gradual change inattenuation would involve the fact that when either increasing or decreasing this inner diameter the two factors determining a, as shown in Equation (1), change in opposite senses.- This: small ,variation in attenuation permits the central conductor to be ;made smaller than :dictated by the optimum eter of the outer conductor and on the frequency, and on the desirability of absolutelyminimum attenuation. Within the range of frequencies and diameters illustrated in Fig. 2, a ten per cent change in diameter ratio occasions less than one-per cent change in attenuation. Wherez-the diameter of the outer conductor is not limited, a diameter ratio of 4.68 will result in anoptimum disposition of a fixed amount of conducting material, as disclosed and claimed in U. S. Patent 1,885,195,- which issued to E.- I. Green on November 1, 1932. Likewise, if for any reason it is desired to useacentral conductor of greater'than optimum "diameter, such may be done with little increase in attenuation. For example, where the overall diameter of the conductor pair is a small fraction of an inch, mechanical considerations might make it desirable to increase the diameter of the central conductor. However, where absolutely minimum.?'ttenuation is desired the outer diameter of the inner conductor should bear the relation to the inner diameter of the outer conductor given'by the equations and drawings disclosed herein.

While specific examples of concentric conductors and of a signalling system embodying the present invention have been described, it is obvious that the invention is' not limited thereto but may find embodiment in other and widely different forms within the scope and spirit of the appended "claims.

What is claimed'is:

1. In a high frequency signaling system, a transmission line comprising a central conductor and an outer cylindrical return conductor concentric therewith andseparated therefrom by a suitable dielectric, the ratio of the outer diameter of said central conductor to the inner diameter of said outer conductor being such that the attenuation of the waves of highest frequency transmitted over said line is a minimum, the inner diameter of said outer conductor and the maximum frequency transmitted over said line being so relatedthat saidoptimum ratio isless than 3.59. I

2. Ina high frequency signaling system, a transmission linecomprising a central conductor and an outer cylindrical return conductorconcentric therewith, means-to apply a wide band of signaling frequencies to said line, the product of the maximum signaling frequency transmitted over said line and the square of the inner diameter of said outer conductor in inches beingless than 10 and the ratio of the inner diameter of said outer conductor to the outer diameter of said central conductor being such that the attenuation of waves of said maximum frequency is a minimum.

3. In a highfrequency signaling system, a transmission line comprising a central conductor and an outer cylindrical return conductor concentric therewith and separated. therefrom by a dielectric whichis chiefly gaseous, the ratio of the inner diameter of said outer conductor to the outer diameter of. said central conductor being not more. than ten per cent greater thanthe ratio for which the attenuation of waves of said maximum frequency in a line having the sameinner'diameter of outer conductor would be a minimum, said latter ratio being lessthan 3.5. V i

4. In a high frequency signaling system, a transmission .line comprising a central conductor and an outer cylindrical return-conductor concentric therewith and separated therefrom Zpk log k j 56 Zp k log k] wherein p-Zvu/ZMXa )\=conductivity of material in abohms per cm.

5. A signaling system comprising in combination means to produce signal waves extending over a wide band of frequencies, and a transmission line associated therewith comprising a central conductor and an outer cylindrical return conductor concentric therewith and separated therefrom by a dielectric which is chiefly gaseous, the ratio of the outer diameter of said central conductor to the inner diameter of said outer conductor being substantially less than 3.59 and so related to said diameter of said outer conductor and to the highest signaling frequency transmitted over said line that the attenuation of the waves of said highest frequency is a minimum.

6. A signaling system in accordance with claim 5 in which the product of the maximum signaling frequency transmitted over said line and the square of the inner diameter of said outer conductor in inches is less than 10 W, In a system adapted to transmit signaling waves of carrier frequencies, a transmission line associated therewith comprising a central conductor and an outer cylindrical conductor concentric therewith and separated therefrom by a dielectric which is chiefly gaseous, the ratio of the inner diameter of said outer conductor to the outer diameter of said central conductor differing by not more than ten per cent from the ratio at which, the inner diameter of said outer conductor being the same, the attenuation of the waves of highest frequency transmitted would be a minimum, said outer conductor being a fraction of an inch in diameter and said highest frequency being so related to the inner diameter of said outer conductor that said last-mentioned ratio is not greater than 3.5.

8. In a signaling system adapted to transmit a maximum frequency of the order of hundreds of kilocycles per second, a transmission line comprising a central conductor and a hollow return conductor concentric therewith, said return conductor being a fraction of an inch in diameter, and the ratio of the inner diameter of said return conductor to the outer diameter of said central conductor being such that the attenuation of waves of said maximum frequency is a minimum.

JULIAN M. WEST. 

